[论文解读] (Sub)linear Kernels for Edge Modification Problems Towards Structured Graph Classes

Bathie, Gabriel, Bousquet, Nicolas+1 more|arXiv (Cornell University)|Jan 1, 2021

Advanced Graph Theory Research参考文献 24被引用 1

一句话总结

本文首次提出了针对图边修改问题的次线性核，具体针对Clique + Independent Set Deletion问题，实现了大小为O(k / log k)的核。该研究提出了一种新颖的Label-And-Reduce核化框架，利用星型森林和中心集的结构性质，安全地减少顶点和边的数量，证明了线性核并非在所有情况下都是最优的，并在指数时间假设（ETH）下建立了紧致的核大小界。

ABSTRACT

In a (parameterized) graph edge modification problem, we are given a graph $G$, an integer $k$ and a (usually well-structured) class of graphs $\mathcal{G}$, and ask whether it is possible to transform $G$ into a graph $G' \in \mathcal{G}$ by adding and/or removing at most $k$ edges. Parameterized graph edge modification problems received considerable attention in the last decades. In this paper, we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if this problem admits a $2k$ kernel [Cao, 2012], this kernel does not reduce the size of most instances. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size $O(k/\log k)$. We also obtain small kernels for several other edge modification problems. We prove that Split Addition (and the equivalent Split Deletion) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. [Ghosh et al., 2015]. We complement this result by proving that Trivially Perfect Addition admits a quadratic kernel (improving the cubic kernel of Guo [Guo, 2007]), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under ETH.

研究动机与目标

• 探究线性核是否为高度结构化图类上边修改问题的理论极限。
• 确定对于目标图为团或独立集划分的边修改问题，次线性核是否可能实现。
• 改进已知问题（如Split Addition、Trivially Perfect Addition和Starforest Deletion）的核大小。
• 在指数时间假设（ETH）下建立紧致的核大小界，证明某些结果的最优性。

提出的方法

• 提出一种Label-And-Reduce核化框架，识别并标记顶点为最优解中星形结构的潜在中心。
• 应用一系列安全约化规则：(1) 移除平凡连通分量（1–2个顶点），(2) 将与度为1的顶点相邻的顶点标记为中心集候选，(3) 移除非中心顶点与中心候选之间的冗余边，(4) 移除中心候选之间的边，(5) 将所有中心候选收缩为单个超顶点，(6) 将度为1的顶点数量限制为k+2。
• 通过星型森林的结构性质分析，界定正实例中度为1的顶点数量，证明m条边意味着至少m−3k个此类顶点。
• 应用最终的核大小规则：若|V(G)| > 4k+3，则返回平凡的否定实例，基于证明表明在此界限之外不可能存在有效解。
• 利用极值图论方法，基于最小度数和环/路径结构，优化核界并改进常数。
• 基于ETH的下界证明表明，Starforest Deletion无法实现次线性核，从而证明线性核的最优性。

实验结果

研究问题

• RQ1 对于目标图为簇图或分裂图等高度结构化图类的边修改问题，能否实现次线性核？
• RQ2 Cluster Editing问题中线性核大小为2k是否为理论极限，还是可通过结构洞察实现更优的核大小？
• RQ3 Label-And-Reduce框架能否推广至其他具有高度结构化目标类的边修改问题？
• RQ4 Starforest Deletion的最优核大小是多少？在ETH下能否实现次线性核？
• RQ5 能否利用星型森林和中心集的结构性质，设计出安全且高效的约化规则？

主要发现

• 本文首次提出了已知的图边修改问题的次线性核：Clique + Independent Set Deletion可实现大小为O(k / log k)的核。
• 证明Split Addition（及Split Deletion）可实现线性核，优于此前已知的二次核。
• 证明Trivially Perfect Addition可实现二次核，优于先前的三次核结果。
• Starforest Deletion可实现大小至多为4k + 3的线性核，且在指数时间假设（ETH）下为最优。
• 本文证明在ETH下，Starforest Deletion不存在次线性核，从而确立了线性核的最优性。
• 基于中心集标记与安全边/顶点约化，Label-And-Reduce核化框架可实现多个问题的紧致核界。